Necking (engineering)

Last updated October 14, 2024

In engineering and materials science, necking is a mode of tensile deformation where relatively large amounts of strain localize disproportionately in a small region of the material. The resulting prominent decrease in local cross-sectional area provides the basis for the name "neck". Because the local strains in the neck are large, necking is often closely associated with yielding, a form of plastic deformation associated with ductile materials, often metals or polymers.^[1] Once necking has begun, the neck becomes the exclusive location of yielding in the material, as the reduced area gives the neck the largest local stress.

Formation

Necking results from an instability during tensile deformation when the cross-sectional area of the sample decreases by a greater proportion than the material strain hardens. Armand Considère published the basic criterion for necking in 1885, in the context of the stability of large scale structures such as bridges.^[2] Three concepts provide the framework for understanding neck formation.

Before deformation, all real materials have heterogeneities such as flaws or local variations in dimensions or composition that cause local fluctuations in stresses and strains. To determine the location of the incipient neck, these fluctuations need only be infinitesimal in magnitude.
During plastic tensile deformation the material decreases in cross-sectional area due to the incompressibility of plastic flow. (Not due to the Poisson effect, which is linked to elastic behaviour.)
During plastic tensile deformation the material strain hardens. The amount of hardening varies with extent of deformation.

The latter two effects determine the stability while the first effect determines the neck's location.

The Considère treatment

Instability (onset of necking) is expected to occur when an increase in the (local) strain produces no net increase in the load, $F$ . This will happen when

\Delta F=0

This leads to

{\begin{aligned}&F=A\sigma _{T}\\[4pt]\implies &dF=Ad\sigma _{T}+\sigma _{T}dA=0\\[4pt]\implies &{\frac {d\sigma _{T}}{\sigma _{T}}}=-{\frac {dA}{A}}={\frac {dL}{L}}=d\varepsilon _{T}\\\implies &\sigma _{T}={\frac {d\sigma _{T}}{d\varepsilon _{T}}}\end{aligned}}

with the $T$ subscript being used to emphasize that these stresses and strains must be true values. Necking is thus predicted to start when the slope of the true stress / true strain curve falls to a value equal to the true stress at that point.

Application to metals

Necking commonly arises in both metals and polymers. However, while the phenomenon is caused by the same basic effect in both materials, they tend to have different types of (true) stress-strain curve, such that they should be considered separately in terms of necking behaviour. For metals, the (true) stress tends to rise monotonically with increasing strain, although the gradient (work hardening rate) tends to fall off progressively. This is primarily due to a progressive fall in dislocation mobility, caused by interactions between them. With polymers, on the other hand, the curve can be more complex. For example, the gradient can in some cases rise sharply with increasing strain, due to the polymer chains becoming aligned as they reorganise during plastic deformation. This can lead to a stable neck. No effect of this type is possible in metals.

The figure shows a screenshot from an interactive simulation available on the DoITPoMS educational website. The construction is shown for a (true) stress-strain curve represented by a simple analytical expression (Ludwik-Hollomon).

The condition can also be expressed in terms of the nominal strain:

{\begin{aligned}{\frac {d\sigma _{T}}{d\varepsilon _{T}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {d\varepsilon _{N}}{d\varepsilon _{T}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {dL/L_{0}}{dL/L}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {L}{L_{0}}}\\[4pt]&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{\mathrm {N} })\end{aligned}}

Therefore, at the instability point:

 $\sigma _{T}={\frac {d\sigma _{T}}{d\varepsilon _{N}}}(1+\varepsilon _{N})$

It can therefore also be formulated in terms of a plot of true stress against nominal strain. On such a plot, necking will start where a line from the point $ε N = -1$ forms a tangent to the curve. This is shown in the next figure, which was obtained using the same Ludwik-Hollomon representation of the true stress – true strain relationship as that of the previous figure.

Importantly, the condition also corresponds to a peak (plateau) in the nominal stress – nominal strain plot. This can be seen on obtaining the gradient of such a plot by differentiating the expression for $σ N$ with respect to $ε N$ .

{\begin{aligned}\sigma _{N}&={\frac {\sigma _{T}}{1+\varepsilon _{N}}}\\[4pt]\therefore {\frac {d\sigma _{N}}{d\varepsilon _{N}}}&={\frac {d\sigma _{T}}{d\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}\end{aligned}}

Substituting for the true stress – nominal strain gradient (at the onset of necking):

{\frac {d\sigma _{N}}{d\varepsilon _{N}}}={\frac {\sigma _{T}}{1+\varepsilon _{N}}}{\frac {1}{1+\varepsilon _{N}}}-{\frac {\sigma _{T}}{(1+\varepsilon _{N})^{2}}}=0

This condition can also be seen in the two figures. Since many stress-strain curves are presented as nominal plots, and this is a simple condition that can be identified by visual inspection, it is in many ways the easiest criterion to use to establish the onset of necking. It also corresponds to the “strength” (ultimate tensile stress), at least for metals that do neck (which covers the majority of “engineering” metals). On the other hand, the peak in a nominal stress-strain curve is commonly a fairly flat plateau, rather than a sharp maximum, so accurate assessment of the strain at the onset of necking may be difficult. Nevertheless, this strain is a meaningful indication of the “ductility” of the metal – more so than the commonly-used “nominal strain at fracture”, which depends on the aspect ratio of the gauge length of the tensile test-piece^[3] – see the article on ductility.

Application to polymers

The tangent construction shown above is rarely used in interpreting the stress-strain curves of metals. However, it is popular for analysis of the tensile drawing of polymers.^[4]^[5] (since it allows study of the regime of stable necking). It may be noted that, for polymers, the strain is commonly expressed as a “draw ratio”, rather than a strain: in this case, extrapolation of the tangent is carried out to a draw ratio of zero, rather than a strain of -1.

The plots relate (top) to a material that forms a stable neck and (bottom) a material that deforms homogeneously at all draw ratios.

As deformation proceeds, the geometric instability causes strain to continue concentrating in the neck until the material either ruptures or the necked material hardens enough, as indicated by the second tangent point in the top diagram, to cause other regions of the material to deform instead. The amount of strain in the stable neck is called the natural draw ratio^[6] because it is determined by the material's hardening characteristics, not the amount of drawing imposed on the material. Ductile polymers often exhibit stable necks because molecular orientation provides a mechanism for hardening that predominates at large strains.^[7]

Related Research Articles

Ductility refers to the ability of a material to sustain significant plastic deformation before fracture. Plastic deformation is the permanent distortion of a material under applied stress, as opposed to elastic deformation, which is reversible upon removing the stress. Ductility is a critical mechanical performance indicator, particularly in applications that require materials to bend, stretch, or deform in other ways without breaking. The extent of ductility can be quantitatively assessed using the percent elongation at break, given by the equation:

A composite material is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or physical properties and are merged to create a material with properties unlike the individual elements. Within the finished structure, the individual elements remain separate and distinct, distinguishing composites from mixtures and solid solutions. Composite materials with more than one distinct layer are called composite laminates.

Young's modulus is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Young's modulus is defined as the ratio of the stress applied to the object and the resulting axial strain in the linear elastic region of the material.

In engineering, deformation may be elastic or plastic. If the deformation is negligible, the object is said to be rigid.

In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and strain can be determined. These curves reveal many of the properties of a material, such as the Young's modulus, the yield strength and the ultimate tensile strength.

<span class="mw-page-title-main">Compressive strength</span> Capacity of a material or structure to withstand loads tending to reduce size

In mechanics, compressive strength is the capacity of a material or structure to withstand loads tending to reduce size (compression). It is opposed to tensile strength which withstands loads tending to elongate, resisting tension. In the study of strength of materials, compressive strength, tensile strength, and shear strength can be analyzed independently.

In materials science, creep is the tendency of a solid material to undergo slow deformation while subject to persistent mechanical stresses. It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increases as they near their melting point.

$<span class="mw-page-title-main">Toughness</span> Material ability to absorb energy and plastically deform without fracturing$

In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing. Toughness is the strength with which the material opposes rupture. One definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing. This measure of toughness is different from that used for fracture toughness, which describes the capacity of materials to resist fracture. Toughness requires a balance of strength and ductility.

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist both shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

Work hardening, also known as strain hardening, is the process by which a material's load-bearing capacity (strength) increases during plastic (permanent) deformation. This characteristic is what sets ductile materials apart from brittle materials. Work hardening may be desirable, undesirable, or inconsequential, depending on the application.

In materials science and engineering, the yield point is the point on a stress–strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible and is known as plastic deformation.

The strain hardening exponent, usually denoted $, is a measured parameter that quantifies the ability of a material to become stronger due to strain hardening. Strain hardening is the process by which a material's load-bearing capacity increases during plastic (permanent) strain, or deformation. This characteristic is what sets ductile materials apart from brittle materials. The uniaxial tension test is the primary experimental method used to directly measure a material's stress-strain behavior, providing valuable insights into its strain-hardening behavior.$

The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 by James R. Rice, who showed that an energetic contour path integral was independent of the path around a crack.

In materials science the flow stress, typically denoted as Y_f, is defined as the instantaneous value of stress required to continue plastically deforming a material - to keep it flowing. It is most commonly, though not exclusively, used in reference to metals. On a stress-strain curve, the flow stress can be found anywhere within the plastic regime; more explicitly, a flow stress can be found for any value of strain between and including yield point and excluding fracture : $.$

The Ramberg–Osgood equation was created to describe the nonlinear relationship between stress and strain—that is, the stress–strain curve—in materials near their yield points. It is especially applicable to metals that harden with plastic deformation, showing a smooth elastic-plastic transition. As it is a phenomenological model, checking the fit of the model with actual experimental data for the particular material of interest is essential.

Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.

Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Depending on the conditions most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile.

<span class="mw-page-title-main">Fiber-reinforced composite</span>

A fiber-reinforced composite (FRC) is a composite building material that consists of three components:

the fibers as the discontinuous or dispersed phase,
the matrix as the continuous phase, and
the fine interphase region, also known as the interface.

In geotechnical engineering, rock mass plasticity is the study of the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture, while plasticity is identified with ductile materials such as metals. In field-scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last word.

Flow plasticity is a solid mechanics theory that is used to describe the plastic behavior of materials. Flow plasticity theories are characterized by the assumption that a flow rule exists that can be used to determine the amount of plastic deformation in the material.

References

↑ Kinloch, AJ; Young, RJ (1995). Fracture Behaviour of Polymers. Chapman and Hall. p. 108. ISBN 9789401715966.
↑ Considère, Armand (1885). Annales des Ponts et Chaussées. 9: 574.{{cite journal}}: Missing or empty |title= (help)
↑ Matic, P; Kirby, GC; Jolles, MI (1988). "The Relation of Tensile Specimen Size and Geometry Effects to Unique Constitutive Parameters for Ductile Materials". Proceedings of the Royal Society of London A. 417 (1853): 309–333. Bibcode:1988RSPSA.417..309M. doi:10.1098/rspa.1988.0063. S2CID 43033448.
↑ McKinley, GH; Hassager, O (1999). "The Considere Condition and Rapid Stretching of Linear and Branched Polymer Melts". Journal of Rheology. 43 (5): 1195–1212. Bibcode:1999JRheo..43.1195M. CiteSeerX 10.1.1.498.6808 . doi:10.1122/1.551034.
↑ Crist, B; Metaxas, C (2004). "Neck Propagation in Polyethylene". Journal of Polymer Science Part B. 42 (11): 2081–2091. Bibcode:2004JPoSB..42.2081C. doi:10.1002/polb.20087.
↑ Séguéla, R (2007). "On the Natural Draw Ratio of Semi-Crystalline Polymers: Review of the Mechanical, Physical and Molecular Aspects". Macromolecular Materials and Engineering. 292 (3): 235–244. doi:10.1002/mame.200600389.
↑ Haward, JN (2007). "Strain Hardening of High Density Polyethylene". Journal of Polymer Science Part B. 45 (9): 1090–1099. Bibcode:2007JPoSB..45.1090H. doi:10.1002/polb.21123.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Kinloch_&_Young-1] Kinloch, AJ; Young, RJ (1995). Fracture Behaviour of Polymers. Chapman and Hall. p. 108. ISBN 9789401715966.

[Considère-2] Considère, Armand (1885). Annales des Ponts et Chaussées. 9: 574.{{cite journal}}: Missing or empty |title= (help)

[3] Matic, P; Kirby, GC; Jolles, MI (1988). "The Relation of Tensile Specimen Size and Geometry Effects to Unique Constitutive Parameters for Ductile Materials". Proceedings of the Royal Society of London A. 417 (1853): 309–333. Bibcode:1988RSPSA.417..309M. doi:10.1098/rspa.1988.0063. S2CID 43033448.

[4] McKinley, GH; Hassager, O (1999). "The Considere Condition and Rapid Stretching of Linear and Branched Polymer Melts". Journal of Rheology. 43 (5): 1195–1212. Bibcode:1999JRheo..43.1195M. CiteSeerX 10.1.1.498.6808 . doi:10.1122/1.551034.

[5] Crist, B; Metaxas, C (2004). "Neck Propagation in Polyethylene". Journal of Polymer Science Part B. 42 (11): 2081–2091. Bibcode:2004JPoSB..42.2081C. doi:10.1002/polb.20087.

[6] Séguéla, R (2007). "On the Natural Draw Ratio of Semi-Crystalline Polymers: Review of the Mechanical, Physical and Molecular Aspects". Macromolecular Materials and Engineering. 292 (3): 235–244. doi:10.1002/mame.200600389.

[7] Haward, JN (2007). "Strain Hardening of High Density Polyethylene". Journal of Polymer Science Part B. 45 (9): 1090–1099. Bibcode:2007JPoSB..45.1090H. doi:10.1002/polb.21123.

[1]

[2]

[3]

[4]

[5]

[6]

[7]